3.729 \(\int \frac{x^3}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=175 \[ \frac{3 \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} d^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} ((5 b c-3 a d) (a d+3 b c)-2 b d x (5 b c-a d))}{4 b^2 d^3 (b c-a d)}-\frac{2 c x^2 \sqrt{a+b x}}{d \sqrt{c+d x} (b c-a d)} \]

[Out]

(-2*c*x^2*Sqrt[a + b*x])/(d*(b*c - a*d)*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c +
d*x]*((5*b*c - 3*a*d)*(3*b*c + a*d) - 2*b*d*(5*b*c - a*d)*x))/(4*b^2*d^3*(b*c -
a*d)) + (3*(5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(S
qrt[b]*Sqrt[c + d*x])])/(4*b^(5/2)*d^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.335447, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{3 \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} d^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} ((5 b c-3 a d) (a d+3 b c)-2 b d x (5 b c-a d))}{4 b^2 d^3 (b c-a d)}-\frac{2 c x^2 \sqrt{a+b x}}{d \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(Sqrt[a + b*x]*(c + d*x)^(3/2)),x]

[Out]

(-2*c*x^2*Sqrt[a + b*x])/(d*(b*c - a*d)*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c +
d*x]*((5*b*c - 3*a*d)*(3*b*c + a*d) - 2*b*d*(5*b*c - a*d)*x))/(4*b^2*d^3*(b*c -
a*d)) + (3*(5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(S
qrt[b]*Sqrt[c + d*x])])/(4*b^(5/2)*d^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.742, size = 165, normalized size = 0.94 \[ \frac{2 c x^{2} \sqrt{a + b x}}{d \sqrt{c + d x} \left (a d - b c\right )} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (- \frac{b d x \left (a d - 5 b c\right )}{2} + \left (\frac{a d}{4} + \frac{3 b c}{4}\right ) \left (3 a d - 5 b c\right )\right )}{b^{2} d^{3} \left (a d - b c\right )} + \frac{3 \left (a^{2} d^{2} + 2 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{5}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(d*x+c)**(3/2)/(b*x+a)**(1/2),x)

[Out]

2*c*x**2*sqrt(a + b*x)/(d*sqrt(c + d*x)*(a*d - b*c)) - sqrt(a + b*x)*sqrt(c + d*
x)*(-b*d*x*(a*d - 5*b*c)/2 + (a*d/4 + 3*b*c/4)*(3*a*d - 5*b*c))/(b**2*d**3*(a*d
- b*c)) + 3*(a**2*d**2 + 2*a*b*c*d + 5*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(s
qrt(b)*sqrt(c + d*x)))/(4*b**(5/2)*d**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.254186, size = 147, normalized size = 0.84 \[ \frac{3 \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{5/2} d^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-\frac{3 a d}{b^2}+\frac{8 c^3}{(c+d x) (a d-b c)}+\frac{2 d x-7 c}{b}\right )}{4 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(Sqrt[a + b*x]*(c + d*x)^(3/2)),x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*((-3*a*d)/b^2 + (8*c^3)/((-(b*c) + a*d)*(c + d*x))
+ (-7*c + 2*d*x)/b))/(4*d^3) + (3*(5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Log[b*c + a*
d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*b^(5/2)*d^(7/2)
)

_______________________________________________________________________________________

Maple [B]  time = 0.042, size = 673, normalized size = 3.9 \[{\frac{1}{ \left ( 8\,ad-8\,bc \right ){b}^{2}{d}^{3}}\sqrt{bx+a} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{3}{d}^{4}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{2}bc{d}^{3}+9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xa{b}^{2}{c}^{2}{d}^{2}-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{3}{c}^{3}d+4\,{x}^{2}ab{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-4\,{x}^{2}{b}^{2}c{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}c{d}^{3}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}b{c}^{2}{d}^{2}+9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{2}{c}^{3}d-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}{c}^{4}-6\,x{a}^{2}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-4\,xabc{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+10\,x{b}^{2}{c}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-6\,{a}^{2}c{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-8\,ab{c}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+30\,{b}^{2}{c}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(d*x+c)^(3/2)/(b*x+a)^(1/2),x)

[Out]

1/8*(b*x+a)^(1/2)*(3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b
*c)/(b*d)^(1/2))*x*a^3*d^4+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/
2)+a*d+b*c)/(b*d)^(1/2))*x*a^2*b*c*d^3+9*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/
2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a*b^2*c^2*d^2-15*ln(1/2*(2*b*d*x+2*((b*x+
a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*b^3*c^3*d+4*x^2*a*b*d^3*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-4*x^2*b^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^
(1/2)+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/
2))*a^3*c*d^3+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(
b*d)^(1/2))*a^2*b*c^2*d^2+9*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2
)+a*d+b*c)/(b*d)^(1/2))*a*b^2*c^3*d-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)
*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^3*c^4-6*x*a^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)-4*x*a*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+10*x*b^2*c^2*d*((b*
x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*a^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-
8*a*b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*b^2*c^3*((b*x+a)*(d*x+c))^(1/
2)*(b*d)^(1/2))/(a*d-b*c)/(b*d)^(1/2)/b^2/((b*x+a)*(d*x+c))^(1/2)/d^3/(d*x+c)^(1
/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(sqrt(b*x + a)*(d*x + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.419415, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (15 \, b^{2} c^{3} - 4 \, a b c^{2} d - 3 \, a^{2} c d^{2} - 2 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} +{\left (5 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (5 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - a^{2} b c^{2} d^{2} - a^{3} c d^{3} +{\left (5 \, b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{16 \,{\left (b^{3} c^{2} d^{3} - a b^{2} c d^{4} +{\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x\right )} \sqrt{b d}}, -\frac{2 \,{\left (15 \, b^{2} c^{3} - 4 \, a b c^{2} d - 3 \, a^{2} c d^{2} - 2 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} +{\left (5 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (5 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - a^{2} b c^{2} d^{2} - a^{3} c d^{3} +{\left (5 \, b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{8 \,{\left (b^{3} c^{2} d^{3} - a b^{2} c d^{4} +{\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(sqrt(b*x + a)*(d*x + c)^(3/2)),x, algorithm="fricas")

[Out]

[-1/16*(4*(15*b^2*c^3 - 4*a*b*c^2*d - 3*a^2*c*d^2 - 2*(b^2*c*d^2 - a*b*d^3)*x^2
+ (5*b^2*c^2*d - 2*a*b*c*d^2 - 3*a^2*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x +
c) - 3*(5*b^3*c^4 - 3*a*b^2*c^3*d - a^2*b*c^2*d^2 - a^3*c*d^3 + (5*b^3*c^3*d - 3
*a*b^2*c^2*d^2 - a^2*b*c*d^3 - a^3*d^4)*x)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^
2)*sqrt(b*x + a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2
+ 8*(b^2*c*d + a*b*d^2)*x)*sqrt(b*d)))/((b^3*c^2*d^3 - a*b^2*c*d^4 + (b^3*c*d^4
- a*b^2*d^5)*x)*sqrt(b*d)), -1/8*(2*(15*b^2*c^3 - 4*a*b*c^2*d - 3*a^2*c*d^2 - 2*
(b^2*c*d^2 - a*b*d^3)*x^2 + (5*b^2*c^2*d - 2*a*b*c*d^2 - 3*a^2*d^3)*x)*sqrt(-b*d
)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(5*b^3*c^4 - 3*a*b^2*c^3*d - a^2*b*c^2*d^2 - a
^3*c*d^3 + (5*b^3*c^3*d - 3*a*b^2*c^2*d^2 - a^2*b*c*d^3 - a^3*d^4)*x)*arctan(1/2
*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/((b^3*c^2*
d^3 - a*b^2*c*d^4 + (b^3*c*d^4 - a*b^2*d^5)*x)*sqrt(-b*d))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(d*x+c)**(3/2)/(b*x+a)**(1/2),x)

[Out]

Integral(x**3/(sqrt(a + b*x)*(c + d*x)**(3/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273937, size = 412, normalized size = 2.35 \[ \frac{{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (b^{6} c d^{4}{\left | b \right |} - a b^{5} d^{5}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{9} c d^{5} - a b^{8} d^{6}} - \frac{5 \, b^{7} c^{2} d^{3}{\left | b \right |} + 2 \, a b^{6} c d^{4}{\left | b \right |} - 7 \, a^{2} b^{5} d^{5}{\left | b \right |}}{b^{9} c d^{5} - a b^{8} d^{6}}\right )} - \frac{15 \, b^{8} c^{3} d^{2}{\left | b \right |} - 9 \, a b^{7} c^{2} d^{3}{\left | b \right |} - 3 \, a^{2} b^{6} c d^{4}{\left | b \right |} + 5 \, a^{3} b^{5} d^{5}{\left | b \right |}}{b^{9} c d^{5} - a b^{8} d^{6}}\right )} \sqrt{b x + a}}{4 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} - \frac{3 \,{\left (5 \, b^{2} c^{2}{\left | b \right |} + 2 \, a b c d{\left | b \right |} + a^{2} d^{2}{\left | b \right |}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt{b d} b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(sqrt(b*x + a)*(d*x + c)^(3/2)),x, algorithm="giac")

[Out]

1/4*((b*x + a)*(2*(b^6*c*d^4*abs(b) - a*b^5*d^5*abs(b))*(b*x + a)/(b^9*c*d^5 - a
*b^8*d^6) - (5*b^7*c^2*d^3*abs(b) + 2*a*b^6*c*d^4*abs(b) - 7*a^2*b^5*d^5*abs(b))
/(b^9*c*d^5 - a*b^8*d^6)) - (15*b^8*c^3*d^2*abs(b) - 9*a*b^7*c^2*d^3*abs(b) - 3*
a^2*b^6*c*d^4*abs(b) + 5*a^3*b^5*d^5*abs(b))/(b^9*c*d^5 - a*b^8*d^6))*sqrt(b*x +
a)/sqrt(b^2*c + (b*x + a)*b*d - a*b*d) - 3/4*(5*b^2*c^2*abs(b) + 2*a*b*c*d*abs(
b) + a^2*d^2*abs(b))*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*
d - a*b*d)))/(sqrt(b*d)*b^3*d^3)